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t 417 ]
XXVI. A new Method of finding the equal Roots of an Equation, by
Dhijion. By the Rev. John Hellins, Curate of Con
ftantine, in Cornwall; communicated by Nevil Maikelyne,
D. D. F. R. S. and AJironomer Royal.
Read June 20, 1783.
^" , ^HE following theorems are a production of juvenile
. years. They were invented about twelve years ago,
when algebra was my favourite ftudy ; and one of them (the
flrfl:) was published as a fpecimen of this method of extracting
'the equal roots of an equation about ten years ago. Since that
time my avocations have left me but very little leifure for im
proving any invention of this kind. Thefe theorems, then,
are in their crude ftate ; however, fuch as they are, I flatter
myfelf,they will afford an eafier folution of equations that have
equal roots than is generally known, and be acceptable to the
ingenious algebraift.
THEOREM I.
If the cubic equation x 3 —px* + qx — r==o has two equal roots s
each of them will be (x) zzJoLzSL
DEMONSTRATION.
Call the three roots #, a, and b; then, by the compofitioa
of equations, we mall have x 3 — 1 ^ x* +™ Ax >~aab 2:04 where
Vol, LXXH. I i i %&
4,i 8 Mr. BELLiNs's M&fyod of finding the
?$Jrhzs.pi r aaJp 2ab~.c[, and aab~r ; which values being
written m our theorem, we have x ( :=. IXZjLi. i zz
v 1pp~bq '
2.aaa + 4,aab + aab+ 2ab,b—qaqir laaq — ^aab^labb @, jr t,
Saai8af+.itib~ baa — izab 9.qa~4qb^2bi> ~~' ' *^ "' :
E.XAME;L:E I.
If the equation x T +5^  32.Y f.36 = has two equal roots^
it is propofed to find them by theabove theorem. ,
H ere p ■==,. — $, q= 32, and, r~ — 36 ; thefe values being
written in the theorem, we have ,~^ x ~3 2 ~9 x ~t . , l6o + 3^
2X25 r 6x32 50+49.%,.
=—=?. 2$ which being written for x, the equation becomes
8+20 — 64 + 36, which is evidently =o; cpnfequently 2 and 2
are roots of it.,
Otherwife, 2, the value of a; given by the theorem, being
written for it in the quadratic equation 3?*+ \ox~..%z = 0, the
refult is 1 2 +20 T..32 ==,©.
Or, dividing the given cubic by, the quadratic #  2I 2 , we,
have x* .4* + 4) x 3 + 5** , 32,* + 36 (* +.9 ; therefore the three.
roots are 2, 2, and —9*
E ; X,.AyM BO^J&vIL
Given x 3 +J2&  g? = 0, aa equation, which.Ms, equal roots^
to find them.
Hereto, and the theorem gives ~: 36poa *4? .~^p whkh
° r 200x9261 ax' vv *" 1 * 1 .
value being written for x the equation vaniflies.
T,;H, ; &0.it.E. M
tqual Roots sf an Epiatim iy^lvpm. 4^
T H EOIEM ».
If the biquadratic equation x*~px s ^qx l >~rx + siZQ has two
equal roots, make A = ^^ '^BPSZ^.Ctt^i, andBzz
, and you will have x
DJ
4A+3/.' J " AC
A fynthetical demonftration of this theorem would be very
long: the investigation is as follows.
It has been demonstrated by the writdrs on algebra, that, if
a biquadratic equation, as x*—px % + qx l ^rx\s~o, has two
equal roots, one of them may be had from the equation
4* 3 — . %px r + zqx  r s o. Multiply this equation by x, and the
original one by 4, and take the difference of the two, which will
be px* — 2^a? z + $rx — 45 =: o. Again, if this equation be multi
plied by 4, and the other cubic by p, and their difference taken >
we mall have 3pp*~ Sq xx* + izr~ zpq x x+pr  i6j , =:o, or
x * + ™pW x+ P>^ :zo or *'+A* + Bito, putting A and
B for the known quantities in the fecond and third terms. Now
multiply this equation by 4.x, and take the firfl cubic from it*
and we (hall have 4A + 3J& xx^ + ^B — zq x x + r— o, which
being divided by 4A + 3/>, and C and D put equal to
$^* nd ^Fy> ref P e & ivel y> gives x* + Cx + t) = o', and
this equation being taken from the other quadratic, there remains
DB
ACx» + BDo; confequently x =fe T— • ^E. I.
corollary i. From the above in vefhigation it appears, that
one of the equal roots may alfo be obtained from either of theife
twoquadratic equations, of which the firft feems moft eligible,
I i i 2 as
420 Mr. itellins's Met bod of finding the
as the coefficients of it are lefs complex than thofe of the:
other ;
$pp — 8q x x*+ I2r ~ 2pq x x +pr ~\ts~ Oy.
and 4A + 3^ x x* + 4B — zq x xhr = o. And thefe*
when p~Oy become  8qx x + 1 %rx — 1 6s — q,
and  5~x ■$<■£ 2a xx + r =a.
Of 5£ ■ *
or X — i#+ — = o>
3* 6
eoROL. 2. If both p and 5 vanifh, then, from either of the
quadratics. we get x=s.~, perfectly agreeing with the cubic
p>x 3 — zqx* +3rx — 45 — o, which, when^ and q vanifh,. becomes
$fx —. 4s =*tO. And this equation, is of ufe ; beeaufe, in this
cafe, the theorem fails,, one of the divifors being — a.
corojl. 3.. From the equation 4.x 3 — $px*+ %qx — rzz o, which*
when p and % vani(h, becomes 4^ 3 r=:o, we alfb get
*.= ~\f  $, another. expreffion, of the fame value of x.
corol. 4. When r = o, D = o, and from the equation
**"+,. C* + D =fco, we have x =.  C.
EX A MP L E> I:
If the equation x**  q#* + 4*+ 12 = has equal roots, ifeis
^fopofed to find them*
Here
equal Roots of an Equation by DiviJtctH. 42:
Here / « o, q =  9, r= ■■■  4, and s — 1 z ; and
A becomes = ll y ~ 4 ~
B
D
, DB
~8x
9
3 :*■
_i6 x
8x
9
__8
3 r
—5
_ 4X T
! +3
8 _„
4
?
4 X
— 2
J,
_4
—8 ~~
3

3
z ^ 3
 + ""
1
^'+32
8 + 33"
So _
25
2
J 4.
which being written for x, the equation becomes. 1636 + 8
+ 12 = 0; therefore 2 is one of the roots.
The fame value of x may be difcovered from either of the
quadratic equations mentioned in corollary 1. The proper values
x>f the coefficients being written in the Srft/ofvthem, it becomes
x z ~.l x — =o, where one value of ~x is — ^— I = 2 . The
3 3 3
other quadratic becomes xf  " x F 2 =0, one of whofe roots is
4. a •■
8
EX A..M.P Iv E H.
It being known that the equation x*  x* ~jx % 4 1 3*  6 = o
has two equal roots, to find them.
Here pi, q= 7, r=. 1& and j~6; and A =
59 59 39 1 39 1 23009' " ^
12800 j D— B 12800 r ,  ,
5io6i' and aT(J = TJboo= '» 0ne of the roots fou § ht 
The
42 2 Mr. helli nS*s Method of finding the
The fame value of x may be found from either of the two
general quadratic equations given in corollary i. From the
fir ft of them we get one value, of x.= 21z — Lil = i , And from
6 59
the other, one value ~ iZ2Z_^I2ii which is alfo— i.
,39 «
E 'X A M P L E III.
Given the equation at 4 — I x + ~~ — o, in which two values
1 2 16
of .v are equal to each other, to find them.
By corollary %, we .have x^= lip^liii ..s= . I . By corol. 2,
jo 2 io a ^ J
3/1 1
T H E O R E M HI.
i/ tbe fur/olid equation x y px* + qx 3  rx % + sx  1 = o has two
roots equal to each other, and you make A =*l¥^L q , B = *P r ~ 2Q!
' 4/>i>l0?' 4pp~lOq*
4pp:ioq> ' S& + 4P* 5 A + 4/>' 5A + 4/.' Au»
^ ^ ATI) » * ^ A^G ' ^^ ^ ~ A^G ' ^ W ^^ 0iW ^ ^ ^?/d/
values of x bez= ~~ •
^ 1 — Cr
The inveftigation of this theorem being altogether fimilar to
that of the laft, it is unneceflary to give it here.
The .difference of equations being taken as in the inveftiga
tion of theorem II. it will appear, that one of the equal roots
may alfo be had from any one of the following five equations, of
which fometimes one, fometimes another, will be the moft
eligible
J.
equal Roots of an Equation by Divifion. 423
i . 5# 4  4^x 3 + $qx % ~ zrx + S — Q.
2, fix* — zqx 3 •{■yx 1 4.SX + 5/ = o,
3. a; 5 + Ax* + Bx + C = o.
4. . # 5 4 D,v* 4 E#  F = 0^
5. x 2 + Gx + H = o.
It is obvious, that, when fi vanifhes, the work will be 'con
siderably fhortened; and when both p and q are wanting,
though the above formula fails, yet the equal root may beealily
obtained from the equation fix" *zqx % + 3 rx 1  ^sx + st = o,
which in, that cafe becomes  $rx x — 4SX 4 5^ = 0. Whenever
s is wanting, F, in the fecond cubic above, will be — o, and
eonfequently x may be found from the quadratic equation
xl 4 Da: 4 E = o. But in any of thefe cafes the equal root may
be found by divifion. However, the operation probably will not,
in general, be fo ihort as extracting the root of the quadratic ; I
will therefore haften to give an. example or two of the u.fe of
the theorem.
E'Xi(4'M PL E: I.
Given x +ar 5 — ** +o , 09433 — O, to find x, two values of it
being equal to each other.;
Hereto, q=i, r=ii, j~q, ./=•=■ .©'09433, and we get
A =  1 5 F — o
B — o G= 0*2231
C — +0*23.58 H~ 0*1241
D= +0*4 I '= 00972
£=0*4238 K = 0*185
and x —■ t~zq — 0*48.
The proper values of the coefficients being written in the
fee equations before. mentioned* and fome of them divided by
the*
424 Mr. hellins's Method of finding the
the coefficient of the higheft power of x, we have thefe four
equations, in each of which one' value of x is one of the equal
ones fougirt :
# 3 # + c6x — 0*4—0.
x z  i'5x** + 0*2358 = 0.
M* + 4.x — 0*4238 ==0.
A; i o , 2 23ix — 01241 =0.
Now the moft eligible equation 4sthe quadratic •#* 404* —
04238 = o, whofe affirmative root is ^04638)  0*2 = 0*48 1 1,
agreeing with the value of x found above, buttrue to two
places lower hr the decimal.
EX AM PL E II.
To find the two equal values of at in the equation
64A* 5 — 2o# 2 4 3 =.0.
The given equation being divided by 64, we have
jc 5 03125^ + 0046875 = ; and then, from the firft of the
five equations given above, we get 5X 4  0*625^ = 0, and# =
^0125 = 0*5. .But from the fecond of the equations juft
mentioned, we have 0*9375;***  0*234375 = o, or x z =
21M1375— 02 c,*and x— */o*2 5 = o c.
From the foregoing few >pages it is evident, that rules may
be made for finding the equal roots of equations of more than
live dimensions by divifion ; but the operations by them will,
in moft cafes, be, long and tedious. <It is obvious, however,
that fuch equations may be depreffed to any dimenlion the alge
braift pleafes.
JL
It has indeed beenfuppofed, that the number of equations
that have equal roots is but fmall, and, confequently, that the
chief
equal Roots of < an liquation by Dh if on. 425'
chief ule of the rules for finding their roots is to get limits
and approximations to the roots of equations in general. That
ufe, it muft be allowed, were it the only one, is fufficie.nt to
pay for inveftigating them. But if the equations that have
equal roots mould hereafter be found not fo few as has been
generally received, then the ufe of the above theorems will
become more extenfive.
I beg leave to add, that this mort effay is but a fmall part of
a, work, in which,, if 1 mould ever have leifure to put a finish
ing hand to it, fcmething more on this fubject may very pro
bably appear. In the mean while, I hope, this little piece willi
be candidly received by thofe who have more leifure and, better
abilities; for ftudies of this kind.
Cbnftantine,
lebraary 9, 1782.
Vol. LXXIi; K" K K.